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Prime Reciprocal Magic Square
A prime reciprocal magic square is a magic square using the decimal digits of the reciprocal of a prime number. Consider a unit fraction, like 1/3 or 1/7. In base ten, the remainder, and so the digits, of 1/3 repeats at once: 0.3333... . However, the remainders of 1/7 repeat over six, or 7−1, digits: 1/7 = 0·142857142857142857... If you examine the multiples of 1/7, you can see that each is a cyclic permutation of these six digits: 1/7 = 0·1 4 2 8 5 7... 2/7 = 0·2 8 5 7 1 4... 3/7 = 0·4 2 8 5 7 1... 4/7 = 0·5 7 1 4 2 8... 5/7 = 0·7 1 4 2 8 5... 6/7 = 0·8 5 7 1 4 2... If the digits are laid out as a square, each row will sum to 1+4+2+8+5+7, or 27, and only slightly less obvious that each ''column'' will also do so, and consequently we have a magic square: 1 4 2 8 5 7 2 8 5 7 1 4 4 2 8 5 7 1 5 7 1 4 2 8 7 1 4 2 8 5 8 5 7 1 4 2 However, neither diagonal sums to 27, but all other prime reciprocals in base ten with maximum period of ''p''−1 produce squa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magic Square
In recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The 'order' of the magic square is the number of integers along one side (''n''), and the constant sum is called the ' magic constant'. If the array includes just the positive integers 1,2,...,n^2, the magic square is said to be 'normal'. Some authors take magic square to mean normal magic square. Magic squares that include repeated entries do not fall under this definition and are referred to as 'trivial'. Some well-known examples, including the Sagrada Família magic square and the Parker square are trivial in this sense. When all the rows and columns but not both diagonals sum to the magic constant this gives a ''semimagic square (sometimes called orthomagic square). The mathematical study of magic squares typically deals with their construction, classification, and enumeration. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reciprocal (mathematics)
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/''b'' is ''b''/''a''. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth (1/5 or 0.2), and the reciprocal of 0.25 is 1 divided by 0.25, or 4. The reciprocal function, the function ''f''(''x'') that maps ''x'' to 1/''x'', is one of the simplest examples of a function which is its own inverse (an involution). Multiplying by a number is the same as dividing by its reciprocal and vice versa. For example, multiplication by 4/5 (or 0.8) will give the same result as division by 5/4 (or 1.25). Therefore, multiplication by a number followed by multiplication by its reciprocal yields the original number (since the product of the number and its reciprocal is 1). The term ''reciprocal' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and \sqrt. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Permutation
In mathematics, and in particular in group theory, a cyclic permutation (or cycle) is a permutation of the elements of some set ''X'' which maps the elements of some subset ''S'' of ''X'' to each other in a cyclic fashion, while fixing (that is, mapping to themselves) all other elements of ''X''. If ''S'' has ''k'' elements, the cycle is called a ''k''-cycle. Cycles are often denoted by the list of their elements enclosed with parentheses, in the order to which they are permuted. For example, given ''X'' = , the permutation (1, 3, 2, 4) that sends 1 to 3, 3 to 2, 2 to 4 and 4 to 1 (so ''S'' = ''X'') is a 4-cycle, and the permutation (1, 3, 2) that sends 1 to 3, 3 to 2, 2 to 1 and 4 to 4 (so ''S'' = and 4 is a fixed element) is a 3-cycle. On the other hand, the permutation that sends 1 to 3, 3 to 1, 2 to 4 and 4 to 2 is not a cyclic permutation because it separately permutes the pairs and . The set ''S'' is called the orbit of the cycle. Every permutation on finitely many elemen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Midy's Theorem
In mathematics, Midy's theorem, named after French mathematician E. Midy, is a statement about the decimal expansion of fractions ''a''/''p'' where ''p'' is a prime and ''a''/''p'' has a repeating decimal expansion with an even period . If the period of the decimal representation of ''a''/''p'' is 2''n'', so that :\frac=0.\overline then the digits in the second half of the repeating decimal period are the 9s complement of the corresponding digits in its first half. In other words, :a_i+a_=9 :a_1\dots a_n+a_\dots a_=10^n-1. For example, :\frac=0.\overline\text076+923=999. :\frac=0.\overline\text05882352+94117647=99999999. Extended Midy's theorem If ''k'' is any divisor of ''h'' (where ''h'' is a number of digits of the period of the decimal expansion of ''a''/''p'' (where ''p'' is again a prime)), then Midy's theorem can be generalised as follows. The extended Midy's theoremBassam Abdul-Baki''Extended Midy's Theorem'' 2005. states that if the repeating portion of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bharati Krishna Tirtha's Vedic Mathematics
''Vedic Mathematics'' is a book written by the Indian monk Bharati Krishna Tirtha, and first published in 1965. It contains a list of mathematical techniques, which were falsely claimed to have been retrieved from the Vedas and containing mathematical knowledge. Krishna Tirtha failed to produce the sources, and scholars unanimously note it to be a mere compendium of tricks for increasing the speed of elementary mathematical calculations sharing no overlap with historical mathematical developments during the Vedic period. However, there has been a proliferation of publications in this area and multiple attempts to integrate the subject into mainstream education by right-wing Hindu nationalist governments. Contents The book contains metaphorical aphorisms in the form of sixteen ''sutras'' and thirteen sub-sutras, which Krishna Tirtha states allude to significant mathematical tools.S. G. Dani (December 2006).Myths and reality : On ‘Vedic mathematics’. :* Originally publis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Number
A cyclic number is an integer for which cyclic permutations of the digits are successive integer multiples of the number. The most widely known is the six-digit number 142857, whose first six integer multiples are :142857 × 1 = 142857 :142857 × 2 = 285714 :142857 × 3 = 428571 :142857 × 4 = 571428 :142857 × 5 = 714285 :142857 × 6 = 857142 Details To qualify as a cyclic number, it is required that consecutive multiples be cyclic permutations. Thus, the number 076923 would not be considered a cyclic number, because even though all cyclic permutations are multiples, they are not consecutive integer multiples: :076923 × 1 = 076923 :076923 × 3 = 230769 :076923 × 4 = 307692 :076923 × 9 = 692307 :076923 × 10 = 769230 :076923 × 12 = 923076 The following trivial cases are typically excluded: #single digits, e.g.: 5 #repeated digits, e.g.: 555 #repeated cyclic numbers, e.g.: 142857142857 If leading zeros are not p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recreational Mathematics
Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research and application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited to being an endeavor for amateurs, many topics in this field require no knowledge of advanced mathematics. Recreational mathematics involves mathematical puzzles and games, often appealing to children and untrained adults, inspiring their further study of the subject. The Mathematical Association of America (MAA) includes recreational mathematics as one of its seventeen Special Interest Groups, commenting: Mathematical competitions (such as those sponsored by mathematical associations) are also categorized under recreational mathematics. Topics Some of the more well-known topics in recreational mathematics are Rubik's Cubes, magic squares, fractals, logic puzzles and mathematical chess problems, but this area of mathematics incl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
